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Hamiltonian system, Scattering theory
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\newcommand {\rmin}{{r_{\rm min}}}
\newcommand {\smin}{s_{\rm min}}
\newcommand {\Vmax}{V_{\rm max}} % maximal value of extremum
\newcommand {\Zmax}{Z_{\rm max}} % maximal modulus of a charge
\newcommand {\thmin}{\Theta_{\rm min}} % non-collinear scattering
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\newcommand {\TR}{{\cal T\!R}} % Time reversal
\newcommand {\IP}{{\cal I\!P}} % Impact parameters
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\begin{document}
\title {Qualitative Aspects of Classical Potential Scattering}
\author{Andreas Knauf\thanks{Max-Planck-Institute
for Mathematics in the Sciences,
Inselstr.\ 22--26, D-04103 Leipzig, Germany.
e-mail: knauf@mis.mpg.de
}}
\date{November 1998}
%
\maketitle
%
\begin{abstract}
We derive criteria for the existence of trapped orbits (orbits which are
scattering in the past and bounded in the future). Such orbits exist if the boundary of Hill's region is non-empty and not homeomorphic to a sphere.
For non-trapping energies we introduce a topological degree which
can be non-trivial for low energies, and for Coulombic and other singular
potentials. A sum of non-trapping potentials of disjoint support
is trapping iff at least two of them have non-trivial degree.
For $d\geq 2$ dimensions the potential vanishes if for any
energy above the non-trapping threshold the classical differential
cross section is a continuous function of the asymptotic directions.
\end{abstract}
%
\section{Introduction}
%
A large part of our knowledge concerning atoms and molecules comes from
scattering experiments.
In the simplest case one scatters particles of definite initial velocity by a
molecule and then observes the final distributions of their directions.
This can be modeled by quantum potential scattering.
The most prominent quantal
phenomenon, namely the resonances of the differential cross section for the
Schr\"{o}dinger equation,
is related to the classical phenomenon of {\em bounded
orbits of positive energy}.
If a potential well of positive minimal height confines a bounded configuration
space region (as it is the case for models of radioactive decay)
then the classical orbits in that region are bounded for all times.
As in this case the bounded orbits form a connected component of the energy
shell, there need be no {\em trapped} orbits (orbits of
positive energy which come from spatial infinity and are
bounded in the future, or vice versa).
Quantum mechanically this then leads to so-called shape resonance poles in the
complex energy plane.
These come exponentially near to the real axis in Planck's constant $\hbar$
\cite{CDKS}.
In this article we are interested in {\em semibounded} trapped orbits.
Although these
are necessarily of Liouville measure zero, they also give rise to quantal
resonances (which, however, may have larger distance
from the real axis and thus correspond to states with
shorter life times \cite{GS}).\\[2mm]
%
In {\bf Sect.\ 2} we introduce some notation, give examples for trapping,
and remark that trapped orbits exist iff there are bounded orbits
in the unbounded component of the energy shell (Prop.\ \ref{prop:two}).
Correspondingly, we derive in {\bf Sect.\ 3} criteria for the existence
for a special class of such bounded orbits. In Thm.\ \ref{thm:rel}
it is shown that such so-called brake orbits exist if some relative homotopy
group of Hill's region w.r.t.\ its boundary is non-trivial.
At least for the physical dimensions $d\leq3$ this is the case iff that
(non-empty) boundary is not homeomorphic to a sphere (Cor.\ \ref{coro}).
After defining the differential cross section in {\bf Sect.\ 4},
we introduce in {\bf Sect.~5} for non-trapping energies a degree
of the scattering map, which turns out to be non-trivial in many cases.
In {\bf Sect.\ 6} we analyze potentials which can be decomposed
into a sum of potentials with disjoint compact supports.
If $n\geq2$ of them have non-trivial degree, then the corresponding energy
is trapping, and orbits, visiting these supports in any prescribed
succession, can be found using
symbolic dynamics (Thm.\ \ref{thm:multi}).
In the final {\bf Sect.\ 7} we consider the differential cross section.
Whereas it is smooth (up to the forward direction) for cases like the
$n$-centre problem with a very complicated dynamics, it is never continuous
for any large energy if $d\geq2$ and for a smooth nonzero potential
(Thm.\ \ref{theo}).
%
\section{Trapped Orbits}
%
Let $V\in C^\infty(M,\bR)$ on configuration space $M:=\bR^d_\q$
be a smooth {\em short-range} potential,
that is, for some $\alpha>1$ the partial derivatives decay at
infinity according to
\beq
\frac{\pa^{n}} {\pa q^{n}}V(\q) = \cO\l(|\q|^{-|n|-\alpha}\ri)
\qquad (n\in \bN_{0}^d),
\Leq{smooth}
with multi-index norm $|n|:=\sum_{l=1}^d |n_l|$.
We denote the Hamiltonian flow generated by the restriction of
\[H:T^* M\ar\bR\qmbox{,}H\pq:=\eh\p^{\,2}+V(\q)\]
to the positive energy part
$P:=\{x\in T^* M\mid H(x)>0\}$ of the phase space
by
\[\Phi^t:P\ar P\qmbox{,}\Phi:\bR\times P\ar
P\qmbox{or}(\p(t,x_0),\q(t,x_0)):=\Phi^t(x_0) ,\]
and the energy shells $H^{-1}(E)$
by $\SuE$. For arbitrary potentials $V$ we set $\Vmax:=\sup_\q V(\q)$.
The phase space $P$ is naturally partitioned into the invariant subsets
%
\begin{definition} \label{defi:bound:scattering} %
{\rm
\begin{eqnarray*}
b^{\pm} &:=& \{x\in P\mid \q\,(\pm\bR^+,x)\mbox{ is bounded }\}
\hspace{5mm}\qmbox{,}b^\pm_E:=b^\pm\cap \SuE\\
b &:=& b^{+}\cap b^{-} \quad\qmbox{(the {\em bound}
states)}\hspace{6.5mm}\qmbox{,}
b_E:=b\cap \SuE\\
s^{\pm} &:=& P \setminus b^{\pm}
\hspace{53.5mm}\qmbox{,} s^\pm_E:=s^\pm\cap \SuE \\
s &:=& s^{+}\cap s^{-} \quad\qmbox{(the {\em scattering} states)}\qmbox{,}
s_E:=s\cap \SuE\\
t &:=& P\setminus(b\cup s) \qmbox{(the {\em trapped} states)}
\hspace{2.5mm}\qmbox{,}
t_E:=t\cap \SuE.
\end{eqnarray*} }
\end{definition}
%
Time reversal $(\p,\q)\mapsto(-\p,\q)$ interchanges $b_E^+$ and $b_E^-$.
It is known (see Hunziker \cite{Hu}) that
\[\lim_{t\ar\pm\infty} |\q(t,x_0)|=\infty \qmbox{iff} x_0\in s^\pm\]
so that these are indeed the $\pm$-scattering states.
By (\ref{smooth}) for any $E>0$ there exists a {\em virial radius}
$\Rvir(E)>0$ for which
\[|V(\q)|0$ a {\em trapping energy} if
\[t_E\equiv(b_E^+\cup b_E^-)\setminus b_E
\equiv (b_E^+\cap s_E^-)\cup (b_E^-\cap s_E^+)\neq\emptyset,\]
and denote the set of trapping energies by $\TE$.
So for these energies there exist {\em trapped} trajectories,
coming from infinity
but bounded in the future, and vice versa.
The complementary set
\[\NT:=\bR^+\setminus\TE\]
of non-trapping energies is known to be open
(see the proof of Prop.\ 2.4.1 of~\cite{DG}).\\[2mm]
%
{\bf Example.} For $d=1$ the set $\TE$ of trapping energies equals the set
$$\l\{E>0\l|\exists q\in\bR: V(q)=E,\ DV(q)=0,\ \sup_{q'\leq q} V(q')=E\
{\rm or}\ \sup_{q'\geq q} V(q')=E\ri.\ri\}$$
of `accessible' critical values, and
$\SuE$ is not connected for $E\in\NT$.
For $d\geq 2$ and a {\em centrally symmetric} ($V(\q)=W(|\q|)$) potential
each of the extrema of $W$ at $q>0$ in the
trapping set $\TE_W$ of $W$ gives rise to an interval
$[W(q),u]\subset \TE_V$ in the
trapping set $\TE_V$ of $V$, and $u>W(q)$ if
the extremum at $q$ is a non-degenerate maximum.
$u$ is of the form $u=W_l(q')$ with $W'_l(q')=0$ and $W''_l(q')=0$,
where
\beq
W_l(r):=W(r)+\frac{l^2}{2r^2}
\Leq{effective}
is the {\em effective potential} .
\\[2mm]
%
For $E>0$ {\em Hill's region}
\[\cR_E :=\{ \q \in M \mid V(\q)\leq E\}\]
need not be connected (since there may be potential pits), but for $d\geq2$
there is precisely one noncompact component $\cR_E^u$
of this set, and the same is true
for the energy shell $\SuE$ projecting to Hill's region. We denote this
component by $\SuE^{\rm u}$.
It may well happen that
\beq
b_E^{\rm u}:=b_E\cap\SuE^{\rm u}\neq \emptyset.
\Leq{beu}
%
{\bf Example.} For centrally symmetric potentials the effective potential
(\ref{effective}) has a positive
local maximum at $r_{\rm max}$ for small values $l>0$ of the angular
momentum parameter, if $W<0$ and $W(r)=\cO(r^{-2-\vep})$.
This then leads to a non-empty set $b_E^{\rm u}=b_E$ of bound states
for the energy $E=W_l(r_{\rm max})>\Vmax=0$.
\medskip
%
\begin{proposition} \label{prop:two}
An energy $E>0$ is non-trapping if and only if $b_E^{\rm u}=\emptyset$.
\end{proposition}
%
{\bf Proof.}
$b_E^+\setminus b_E$ lies in $\SuE^{\rm u}$. So if the closed,
$\Phi^t$-invariant set $b_E^+\cap\SuE^{\rm u}$ is non-empty, then the set
of its
$\omega$-limit points lying in the compact region of the energy shell over
$B(E)$ is non-empty, too (see also Prop.\ 2.1.2 of~\cite{DG}).
Thus $b_E^{\rm u}\neq\emptyset$.
To show the inverse implication, we assume that $E>0$ is non-trapping, so that
$t_E=\emptyset$.
Then
\[\SuE^{\rm u}=s_E\,\dot{\cup}\, b_E^{\rm u}\]
so that for $b_E^{\rm u}\neq \emptyset$ there would be a sequence $x_i\in s_E$
of points on scattering
orbits converging to $x:=\lim_{i\ar\infty} x_i\in b_E^{\rm u}$.
Then there exist unique times $t_i$ such that $y_i\equiv(\p_i,\q_i):=\Phi^{t_i}(x_i)$ enter the interaction zone, i.e.\
meet $|\q_i|= \Rvir(E)$ and $\LA \p_i,\q_i\RA\leq c<0$.
By compactness there exists an accumulation point $y\equiv(\p,\q)$ of the $y_i$.
Since $|\q|= \Rvir(E)$ and $\LA \p,\q\RA\leq c$, it is backward scattering
($y\in s_E^-)$.
But the times $t_i\nearrow\infty$, so that $y\in b_E^+$, too.
Thus $y$ belongs to a trapped orbit.
\hfill$\Box$\\[2mm]
%
The virial identity (\ref{virial})
implies that the motion is {\em non-trapping} above some
(optimal) energy threshold $E_{\NT}\geq\Vmax$, i.e.\
\[ ]E_{\NT},\infty[\ \subset \NT \qmbox{and}E_{\NT}\in\TE\mbox{ or }E_{\NT}=0,\]
since for $E$ large
$\frac{d}{dt} \LA\q(t),\p(t) \RA > E$ for all $\q\in\bR_\q^d$.
This implies a unique minimum of $t\mapsto |\q(t,x_0)|$ at, say $t=0$, and
the estimate
\beq
\q^{\,2}(t,x_0)\geq \q_0^{\,2}+Et^2\qquad (t\in\bR).
\Leq{away}
%
\begin{remark} {\rm
Without a smoothness assumption for the potential $V$ this need
not be true even if $V<0$.
Namely, for the physically important {\em $n$-centre potentials} of the form
\beq
V(\q)= \sum_{l=1}^{n}\frac{-Z_{l}}{\B{\q-\s_{l}}} ,
\Leq{n:centers}
one has for $n\geq2$ in $d=2$ dimensions $\TE=\bR^+$, at least if all
charges $Z_l>0$, see \cite{KK}.
For $n\geq2$, $d=3$ and arbitrary $Z_l\neq0$,
the set $\TE$ of trapping energies contains
an interval $[\Eth,\infty[$, see \cite{Kn2}.
However, $\NT\neq\emptyset$, too if all charges $Z_l$ are negative, since then
the radial component of the force $-\nabla V$ is positive outside a ball
containing all $\s_l$, and since for small $E>0$ Hill's region does not contain that ball.
}\end{remark}
%
As this example shows, non-trapping energies can lie below, not only above,
trapping energies. \\[2mm]
%
{\bf Example.} In the smooth case
for $d=1$ the threshold energy is $E_{\NT} = \Vmax$, and
for $E > E_\NT$ all scattering is in
the forward direction.
%In particular the scattering cross section is trivially smooth
%This is not the case for larger dimensions $d$:
%
\section{Brake Orbits}
%
We saw in Prop.\ \ref{prop:two} that $E\in\NT$ iff $b_E^u=\emptyset$.
Here we derive a criterion for the existence of bound states $b_E^u$
for energies $E\leq \Vmax$.
The set $\TE$ of trapping energies contains all critical values
$E$ of $V$ with critical points $\q\in\pa\cR_E^u$, since then
the phase space point $(\vec{0},\q)\in\SuE^u$ belongs to the set $b_E^u$
defined in (\ref{beu}).
So we may ask ourselves whether a regular value $E0$, so that
$\pa B(E)$ is convex in the Jacobi metric.
Now for the first $k\geq 1$ with nontrivial $\pi_k(\cR_E^u,\pa\cR_E^u)$
we consider an essential map $f_0:(B^k,\pa B^k)\mapsto (\cR_E^u,\pa\cR_E^u)$.
We then apply to $f_0$ a curve shortening process, originally
devised by Seifert in \cite{Se}
and used by Gluck and Ziller in \cite{GZ}. Here one considers
$f_0$ as a $(k-1)$-parameter family of curves whose ends lie in $\pa\cR_E^u$.
This is possible since $\pa\cR_E^u$ is compact,
$\nabla V\neq \vec{0}$ on $\pa\cR_E^u$, and
so one may apply to $\pa\cR_E^u$ the metric surgery described
in \cite{GZ}.
Although $\cR_E^u$ is not compact, a Palais-Smale condition holds:
by convexity of $\pa B(E)$ w.r.t.\ $g_E$ the curve shortening leads to
curves still lying inside $B(E)$. Alternatively one may choose the radius
$\Rvir(E)$
of $B(E)$ so large that the $g_E$-distance between $\pa\cR_E^u$ and
$\pa B(E)$ is larger than the maximal length of a curve in the family $f_0$.
So the shortening process leads to a non-trivial geodesic segment
with two end points in $\pa\cR_E^u$. This corresponds to a
periodic brake orbit of energy $E$.
\hfill$\Box$\\[2mm]
%
In fact this criterion is often met:
%
\begin{corollary} \label{coro}
If for $d\leq 3$ the boundary $\pa\cR_E^u$ of Hill's region
is not empty or homeomorphic to $S^{d-1}$, then
there exists a periodic brake orbit in $\SuE^{\rm u}$, and thus
$E\in\TE$.
\end{corollary}
%
{\bf Proof.}
By a remark at the beginning of this section we may again assume that
$E$ is a regular value of $V\rstr_{\cR_E^u}$, so that
$\cR_E^u$ is a smooth $d$-manifold with boundary.
For $d=1$ then one only has the alternatives $\pa\cR_E^u=\emptyset$ or
$\pa\cR_E^u\cong S^0$. So assume $d\geq 2$, and denote by $\ov{\cR_E^u}$ the
compact manifold which arises from $\cR_E^u\subset M=\bR^d$ by the
one-point compactification of $\bR^d$. Now $\ov{\cR_E^u}$ is a compact
manifold with boundary $\pa\ov{\cR_E^u}=\pa\cR_E^u$.
Thus not all relative {\em homology} groups
$H_k(\ov{\cR_E^u},\pa \ov{\cR_E^u})$, $k=1,\ldots,n$,
are trivial (cf.\ Spanier \cite{Sp}, Chapter 4).
For $d\geq2$ Hill's region $\cR_E^u$ and thus also $\ov{\cR_E^u}$ is
connected. We may
assume that $\pa\ov{\cR_E^u}=\pa\cR_E^u$ is connected, too, since otherwise
$\pi_1(\cR_E^u,\pa\cR_E^u)$ is non-trivial and we can apply
Theorem~\ref{thm:rel}. This already shows our claim for $d=2$, since the only
closed connected (non-empty) 1-manifold is $S^1$.
For $d\geq3$ we can apply the relative
Hurewitz isomorphism theorem (\cite{Sp}, Chapter 7.5) to show that
there exists a nontrivial relative homotopy group
$\pi_k(\ov{\cR_E^u},\pa\ov{\cR_E^u})$.
Let $k$ be the smallest such integer. If $k0\}.\]
More precisely, the {\em M\o ller transformations}
\[\Opm := \lim_{t\ar\pm\infty} \Phi^{-t}\circ\Pit\]
exist (pointwisely) on $\Pin$, and
are symplectic diffeomorphisms onto their images $s^{\pm}$, see \cite{Sim}.
In particular the {\em asymptotic momentum}
\[\p^\pm:s^\pm\ar\bR^d\qmbox{,} \p^\pm(x_0):=\lim_{t\ar\pm\infty} \p(t,x_0),\]
the {\em asymptotic direction}
\[\hat{p}^\pm:s^\pm\ar S^{d-1}\qmbox{,}
\hat{p}^\pm(x) := \frac{\p^\pm(x)}{|\p^\pm(x)|}\]
and the {\em impact parameter}
\[\q_\perp^{\,\pm}:s^\pm\ar\bR^d,\quad
\q_\perp^{\,\pm}(x_0) := \lim_{t\ar\pm\infty}
\l(\q(t,x_0)-\LA\q(t,x_0),\hat{p}^\pm(x_0)\RA \hat{p}^\pm(x_0)\ri)\]
are smooth $\Phi^t$-invariant functions.
The impact parameter
is orthogonal to the asymptotic direction, and for $E>0$
\[A_E^\pm :s_E^\pm/\bR\ar I_E^\pm\subset T^*S^{d-1}\qmbox{,}
x\mapsto (\q_\perp^{\,\pm}(x),\hat{p}^\pm(x)) \]
is a homeomorphism onto its (open and dense) image $I_E^\pm$.
Note in comparison to the inverse M\o ller transformations
$\Opm_*: s^\pm\ar \Pin$, that the energy now appears as a parameter, and
that orbits are mapped to points, so that we disregard time delay etc.
For $\hat{I}_E^\pm:=A_E^\pm(s_E/\bR)$
the energy $E$ {\em scattering map}
\beq
(\Q_E,\hat{P}_E): \hat{I}_E^-\ar \hat{I}_E^+\qmbox{,}
(\q_\perp^{\,-},\hat{p}^-)\mapsto
A_E^+\circ (A_E^-)^{-1}(\q_\perp^{\,-},\hat{p}^-)
\Leq{sc:map}
from the initial to the final asymptotic data is a symplectic diffeomorphism
w.r.t.\ the canonical symplectic form $\omega_N$ on the cotangent bundle
\[N:=T^*S^{d-1}\]
of the sphere of directions.
In particular it preserves the Liouville measure
\[\lambda_N:=\frac{\omega_N\wedge\ldots\wedge\omega_N}{(d-1)!}\]
on $N$.
The differential cross section
$\DCSP$ is the (density of the) number of particles
per second scattered in the final direction
$\htheta^+\in S^{d-1}$, assuming a uniform flux of one particle
per second and unit area of incoming particles of energy $E$ and
initial direction $\htheta^-\in S^{d-1}$.
So we consider the restriction
\beq
\hat{P}_{E,\htheta^-} := \hat{P}_E\rstr_{\hat{I}_{E,\htheta^-}^-}
\Leq{achteinhalb}
of the {\em final direction map} $\hat{P}_E$ to the intersection
\[\hat{I}_{E,\htheta^-}^- := \hat{I}_E^- \cap T^*_{\htheta^-}S^{d-1}\]
of its domain with the cotangent space of the sphere at ${\htheta^-}$.
%
\begin{definition}
For $E>0$ and $\htheta^{-}\in S^{d-1}$
the {\bf cross section measure} $\set$ on $S^{d-1}$ is the image measure
\beq
\set := \l(\hat{P}_{E,\htheta^-}\ri)^{-1}\l( \lambda_{\htheta^{-}} \ri),
\Leq{csm}
$\lambda_{\htheta^{-}}$ being Lebesgue measure on
the cotangent plane $T^*_{\htheta^{-}}S^{d-1}$.
If $\set$ on $S^{d-1}\setminus \{\htheta^{-}\}$ is
absolutely continuous w.r.t.\ Lebesgue measure $\lambda_{S^{d-1}}$,
the Radon-Nikodym derivative $\DCSP$ is called
the {\bf differential cross section}.
\end{definition}
%
If the set $\IP:= \hat{P}_{E,\htheta^-}^{-1}(\htheta^{+})$ of
initial impact parameters
is countable, we may thus write the differential cross section as the sum
\[\DCSP=\sum_{\q_\perp^{\,-}\in \IP}
\l|D\hat{P}_{E,\htheta^-}(\q_\perp^{\,-})\ri|^{-1}.\]
%
{\bf Example.}
For the Coulomb potential $V(\q)=Z/|\q|$, $Z\neq0$ on $\bR^d\setminus\{0\}$
one has the so-called {\em Rutherford} differential cross section
\beq
\DCSP= \l(\frac{|Z|}{4E\sin^2(\eh\sphericalangle(\htheta^+,\htheta^-))}\ri)^{d-1}.
\Leq{Ruth}
%
\section{The Degree of the Scattering Map}
%
For non-trapping energies $E\in\NT$ the scattering map (\ref{sc:map}) is a
symplectic diffeomorphism
\[(\Q_E,\hat{P}_E):N\ar N\]
of $(N,\omega_N)$, and for each $\htheta^-\in S^{d-1}$ the
restriction (\ref{achteinhalb})
\[\hat{P}_{E,\htheta^-}: T^*_{\htheta^-}S^{d-1}\ar S^{d-1}\]
of the final direction map is smooth. For $d\geq 2$
\[\lim_{\q_\perp^{\,-}\ar\infty}\hat{P}_{E,\htheta^-}(\q_\perp^{\,-}) =
\htheta^-.\]
Thus we may extend it uniquely to a continuous map
\beq
\hat{\bf P}_{E,\htheta^-}:
\l( T^*_{\htheta^-}S^{d-1}\cup\{\infty\} \ri) \cong S^{d-1}\ar S^{d-1}.
\Leq{P:E:theta}
The choice of an orientation on the sphere
fixes an orientation of the cotangent space $T^*_{\htheta^-}S^{d-1}$, too,
and we denote by
\[{\rm deg}(E):= {\rm deg}(\hat{\bf P}_{E,\htheta^-})\]
the topological degree of this map (see, e.g., Hirsch \cite{Hi}).
That degree is independent of the choice of orientation.
By continuity of
the final direction map $\hat{P}_{E}$ it is independent of the choice of
initial direction $\htheta^-$.
Furthermore, $\hat{P}_{E}$ depends continuously on $E\in\NT$, so that the
{\em non-trapping degree}
\[{\rm deg}:\NT\ar\bZ\]
is locally constant on the (open) set of non-trapping energies.
Now we will work out a series of examples.
%
\begin{proposition}
For a smooth short-range potential $V$
\[{\rm deg}(E)=0\qquad (E>E_\NT).\]
\end{proposition}
%
{\bf Proof.}
This is obvious for large energies $E$, since then
the map $\hat{\bf P}_{E,\htheta^-}$ is not onto $S^{d-1}$:
The curvature $k$
of the trajectory, that is, the inverse radius of the osculating circle,
can be considered as a phase space function, and equals
\[k:P\setminus \l(\{\vec{0}\}\times M\ri) \ar[0,\infty[\qmbox{,}
k(\p,\q):=\frac{\l|(\idty-\Pi_\p)\ddot{q}\ri|}{|\dot{q}|^2},\]
where $\Pi_\p$ denotes the orthogonal projection in the direction of $\p$.
Inserting Hamilton's equation, we see that
\beq
k(\p,\q)= \frac{\l|(\idty-\Pi_\p)\nabla V(\q)\ri|}{2(E-V(\q))} \leq
\frac{\l|\nabla V(\q)\ri|}{2(E-V(\q))}.
\Leq{geo:curv}
For large $E$ by (\ref{smooth})
the integral of (\ref{geo:curv}) is integrable and, using (\ref{away}),
is seen to be uniformly of order
\[\int_\bR k\circ\Phi^t(x_0)\,|d\q(t,x_0)/dt|\, dt = \cO(H(x_0)^{-1}).\]
This implies absence of back-scattering for large $E$ and thus
${\rm deg}(E)=0$. As the degree is locally constant, the result follows for all
$E\in ]E_\NT,\infty[$. \hfill$\Box$\\[2mm]
%
The following proposition generalizes the case
of the Kepler potential (which corresponds to $n=1$).
%
\begin{proposition} \label{deg:sing}
For $d>1$ let $\Muh:= \bR^d\setminus\{\vec{0}\}$. Then
for $n\in\bN$, the flow generated by the potential
\beq
V(\q) := -|\q|^{-2n/(n+1)}\qquad (\q\in\Muh)
\Leq{V:form}
can be regularized, all positive energies are non-trapping
($\NT=\bR^+$), and the degree of the scattering map equals
\beq
{\rm deg}(E) = \l\{
\begin{array}{cl}
-n& d \ {\rm even}\\
\eh(1-(-1)^n)& d \ {\rm odd}\end{array}\ri.\qquad (E>0).
\Leq{sing:deg}
\end{proposition}
%
{\bf Proof.}
Due to the singularity at the origin the Hamiltonian flow in the phase space
$T^*\Muh$ is incomplete.
We will show, however, that this flow can be completed in an essentially unique
way.
To that aim we calculate the total deflection angle $\Delta\vv(E,l)$
of a trajectory with energy $E$ and modulus $l$ of the angular momentum.
Considering for a moment an arbitrary centrally symmetric potential
$V(\q)=W(|\q|)$
and for $l>0$ its effective potential $W_l$ (see (\ref{effective})),
we have (see Chapter 2.8 of Arnold \cite{Ar})
\beq
\Delta\vv(E,l)=2\int_\rmin^\infty \frac{\dot{\vv}}{\dot{r}}\, dr -\pi=
2\int_\rmin^\infty \frac{l/r^2}{\sqrt{2(E-W_l(r))}}\, dr -\pi,
\Leq{Delta:phi}
where the pericentral radius $\rmin$ is the largest $r>0$ with $W_l(r)=E$.
Setting $W(r):= -r^{-\alpha}$ with $00).\]
For $d>2$ we consider a family of trajectories with fixed $E$ and $\htheta^-$,
whose impact parameter $\q_\perp$ varies on a one-dimensional subspace
$L\subset T^*_{\htheta^-}S^{d-1}$.
$\htheta^-$ and this subspace span a 2--plane in $\bR^d$, and
$\htheta^+$ lies in that plane. To avoid degeneracies we choose a
$\htheta^+$ which is linear independent from $\htheta^-$. Then there are
exactly $n$ impact parameters $\q^1_\perp,\ldots,\q^n_\perp\in L$
with $\hat{{\bf P}}_{E,\htheta^-}(\q^i_\perp)=\htheta^+$.
$[n/2]$ of them have a scalar product $\LA \q^i_\perp, \htheta^+ \RA>0$,
and $\LA \q^i_\perp, \htheta^+ \RA<0$ for the rest.
For the first group the restriction of the linearization of the final angle map
to the subspace $\{\vec{v}\in T^*_{\htheta^-}S^{d-1}\mid \vec{v}\perp L\}$
gives a positive sub-determinant, whereas for the second group
the sign equals $(-1)^{d-2}$. So
\[{\rm deg}(E)=-\l([n/2]+(-1)^{d-2}(n-[n/2])\ri), \]
proving (\ref{sing:deg}).
\hfill$\Box$
%
%
\begin{proposition}
For a centrally symmetric short-range potential $V$
\[{\rm deg}(E)=+1\qmbox{if} E\in\NT\cap]0,\Vmax[.\]
\end{proposition}
%
{\bf Proof.}
When we substitute $v:=r_{\rm min}/r$
in the formula (\ref{Delta:phi}) for the deflection angle, we get
\[\Delta\vv(E,l)=2\int_0^1 \frac{dv}{\sqrt{2r^2_{\rm min}(E-V(r_{\rm
min}/v))/l^2 -v^2}}-\pi.\]
For $d=2$ the degree equals
\beqn
{\rm deg}(E)&=& -\frac{2}{\pi}\int_0^\infty \frac{\pa}{\pa l}\Delta\vv(E,l)dl
\label{interchange}\\
&=& -\frac{2}{\pi}\l.\int_0^1 \frac{dv}{\sqrt{2(r_{\rm min}/l)^2(E-V(r_{\rm
min}/v)) -v^2}}\ri|_{l=0}^{l=\infty}\NN\\
&=& \frac{2}{\pi}\int_0^1\frac{dv}{\sqrt{1-v^2}}=1,\NN
\eeqn
since $\lim_{l\ar\infty} r_{\rm min}(E,l)/l=1/\sqrt{2E}$ and
$\lim_{l\ar 0} r_{\rm min}(E,l)/l>0$, using the assumption $E2$ is treated similar as in
Prop.~\ref{deg:sing}.
\hfill$\Box$\\[2mm]
%
I conjecture that for the above energy range the degree equals one, even if
the potential is not centrally symmetric.
%
\section{Multiple Scattering}
%
We now consider potentials $V\in C^\infty_0(M,\bR)$, $d\geq 2$, whose
support is contained in the union of $n$ disjoint balls
\[B_l:=\{\q\in M\mid |\q-\s_l|\leq r_l\}\qquad(l=1,\ldots,n),\]
and represent $V$ in the form $V=\sum_{l=1}^n V_l$ with
${\rm supp}(V_l)\subset B_l$.
Our aim is to compare the flow $\Pt$ generated by $H$ with the flows $\Pt_l$
generated by the Hamiltonian functions $H_l:P\ar\bR$, where
$H_l(\p,\q):=\eh\p^{\,2}+V_l(\q)$. In general objects corresponding to $V_l$
will carry a subindex $l$.
For $E>0$ we have
\[\cR_E^u=\bigcap_{l=1}^n \cR_{l,E}^u\qmbox{and}
b_E^u\supset\bigcup_{l=1}^n b_{l,E}^u,\]
since $d\geq2$ and the supports of the $V_l$ are disjoint.
So by Prop.\ \ref{prop:two} the set $\NT$ of non-trapping energies of $H$
meets
\[\NT\subset\bigcap_{l=1}^n\NT_l.\]
We now assume that $V$ is {\em non-shadowing}, by which we mean
that every straight line in $M$ meets at most two balls $B_l$.
Moreover, we only consider scattering from and to directions in which the
balls do not shadow each other.
We thus exclude the cones of angles
\beq
\alpha_{k,l}:=\arcsin\l(\frac{r_k+r_l}{d_{k,l}}\ri)\qmbox{with}
d_{k,l}:=|\s_k-\s_l|
\Leq{def:alpha}
around the axes $\hat{s}_{k,l}:=(\s_k-\s_l)/d_{k,l}$,
and restrict the initial and final directions $\hat{p}^\pm$ to the subset
\beq
\tilde{S}^{d-1}:= \{\hat{x}\in S^{d-1}\mid \sphericalangle(\hat{x},\hat{s}_{k,l})>
\alpha_{k,l},\, 1\leq k\neq l\leq n\}
\Leq{tilde:S}
of the sphere not contained in any such cone.
In order to use symbolic dynamics, we introduce
{\em symbol sequences}
\[\uk =(k_i)_{i\in I}\in \cS^I
\qmbox{over the {\em alphabet}} \cS := \{1,\ldots,n\},
\]
where
\[I\equiv I_l^r:=\{i\in\bZ\mid l \leq i \leq r \}\]
for $l,r\in\bZ\cup\{\pm\infty\}$ is a
(finite, half-infinite or bi-infinite) {\em interval}.
$\uk$ is called {\em admissible} if
$k_i\neq k_{i+1}$ for all $\{i,i+1\}\subset I$, and
\[\Adm_l^r := \{\uk\in\cS^I\mid \uk\,\,{\rm admissible}\}.\]
%
\begin{theorem} \label{thm:multi}
Let $n\geq 2$, $E$ be non-trapping for the individual potentials $V_l$
($E\in\cap_{l=1}^n\NT_l$) and
${\rm deg}_l(E)\neq 0$, $1\leq l\leq n$.
Then for every interval $I_l^r$, $\uk\in\Adm_l^r$ and
$\hat{p}^\pm\in\tilde{S}^{d-1}$
there is a trajectory in $\SuE$
meeting exactly the balls $B_{k_i}$, $i\in I_l^r$, in succession.
\begin{itemize}
\item
If $l\neq -\infty$, then this trajectory in $s_E^-$ has initial direction
$\hat{p}^-$. Otherwise it belongs to $b_E^-$.
\item
If $r\neq \infty$, then this trajectory in $s_E^+$ has final direction
$\hat{p}^+$. Otherwise it belongs to $b_E^+$.
\end{itemize}
In particular $E$ is a trapping energy for $V$ ($E\in\TE$).
\end{theorem}
%
{\bf Proof.}
We only need to consider the case $l=10\mid \Pt(x)\in U\}\qquad (x\in \intV\,)\]
is finite, and smooth on $\intV$.
The {\em interior Poincar\'{e} map}
\[\cP^i:V\ar U\qmbox{,} x\mapsto \Phi(T^i(x),x)\]
is a diffeomorphism: \\
By transversality its restriction to $\intV\,$ is a
diffeomorphism, and its restriction to $V\cap U$ equals the identity.
Finally, $\cP^i$ is also smooth at the boundary of its domain.
Namely, by enlarging the balls $B_k$ a bit
(without loosing the non-shadowing property), we may assume that
${\rm supp}(V_k)\cap\pa B_k=\emptyset$, so that the dynamics near the boundary
is the free dynamics.
Thus near the component
$\pa\cD_k^0$ of $V\cap U$
the interior Poincar\'{e} map acquires the smooth form
\[\cP^i(\p,\q)=(\p,\q-2\LA \q-\s_k,\hat{p}\RA\cdot\hat{p}),
\qmbox{with}\hat{p}:=\p/|\p| .\]
On $V(k)$ and on $U(k)$ we use the smooth coordinates
\[(\q^\perp_k,\hat{p}) \qmbox{with}\q^\perp_k:= (\idty-\Pi_\p)(\q-\s_k)\]
($\Pi_\p$ being the $\p$--projection), which map $V(k)$ resp.\ $U(k)$
homeomorphically onto the disk bundle
\[B^kS^{d-1}:= \l\{(\vec{v},\htheta)\in T^*S^{d-1}\mid |\vec{v}|\leq r_k\ri\},\]
and $\intV(k)$ resp.\ $\intU(k)$ diffeomorphically onto the interior.
When we write $\cP^i(k)=(\Q^\perp_k,\hat{P}_k)$, then for a given incoming
direction $\hat{p}\in S^{d-1}$ the map
\beq
B^k_{\hat{p}}S^{d-1}\ni \q^\perp\mapsto \hat{P}_k(\q^\perp,\hat{p})\in S^{d-1}
\Leq{E}
sends the points $\q^\perp$ of modulus $r_k$ onto $\hat{p}$ and thus
can be considered as a map
\beq
\hat{\bf P}_{k,\hat{p}}:S^{d-1}\ar S^{d-1}\qmbox{from the $(d-1)$-sphere}
B^k_{\hat{p}}S^{d-1}/\sim \ \ \cong S^{d-1}
\Leq{I}
to the $(d-1)$--sphere of outgoing directions.
Here $\sim$ identifies the points
$\q^\perp\in B^k_{\hat{p}}S^{d-1}$ of modulus $r_k$.
The trajectories of $\Pt_k$ which do not meet $B_k$ are
straight lines. So the degree of the continuous map $\hat{\bf P}_{k,\hat{p}}$
equals the degree
${\rm deg}_k(E)$ which is non-zero by assumption.
In particular we see that for $r=1$ there is a trajectory
with initial resp.\ final directions $\hat{p}^-,\hat{p}^+$
meeting only $B_{k_1}$. So assume from now on $r\geq2$.\\
%
{\bf 3)}
Since the motion outside the balls $B_k$ is free,
the {\em exterior return time}
\beq
T^e:U\ar\bR\cup\{\infty\}\qmbox{,}
T^e(x) := \inf \l\{ t>0 \l| \Phi^t(x)\in V \ri. \ri\},
\Leq{return:time}
is bounded below by the
minimal distance between the balls, divided by the speed $\sqrt{2E}$.
Due to our non-shadowing assumption, on $U':= \{x\in U\mid T^e(x)0$, see (\ref{def:alpha}), and the rotation
$\cM\equiv\cM(\hat{p}_l,\hat{s})\in{\rm SO}(d)$
in the plane spanned by $\hat{s}$ and $\hat{p}_l$, given by
$$\cM(\vec{v}) := \vec{v}+\frac{
\hat{s}((1+2c)\LA\hat{p}_l,\vec{v}\RA - \LA\hat{s},\vec{v}\RA)
- \hat{p}_l\LA\hat{s} + \hat{p}_l,\vec{v}\RA}{1+c}
\qquad(\vec{v}\in\bR^d)$$
is well defined. $\cM$ maps $\hat{p}_l$ to $\hat{s}$.
The one-parameter family of rotations $\cM_t$ on $\bR^d$
\[\cM_t := \exp(t\log(\cM_1))\qquad (t\in[0,1])\]
is well-defined and smooth in $t$ and $\hat{p}_l$,
since $\cM = \cM_1$ rotates by an angle $0$. As shown in \cite{KK}, the same is true for the
$n$-centre potential (\ref{n:centers}) in $d=2$ dimensions,
although this Hamiltonian system is
non-integrable for $n\geq3$. See \cite{Kn2} for similar results in
$d=3$ dimensions.
On the other hand, the differential cross section is smooth (again, up
to the forward direction) for many smooth potentials $V$ and energies
below $\Vmax$.
So the next result may be unexpected.
%
\begin{theorem} \label{theo}
Let $d\geq 2$ and $V$ a smooth short-range potential of decay
rate $\alpha=2(d-1)$ in (\ref{smooth}).
If the differential cross section
\[ (\htheta^-,\htheta^+)\mapsto \DCSP\]
is continuous on $\l(S^{d-1}\times S^{d-1}\setminus {\rm Diag}\ri)$
for any non-trapping energy $E>E_{\NT}$, then $V\equiv 0$.
\end{theorem}
%
{\bf Proof.}
For $E>\Vmax$ the configuration space
trajectories $t\mapsto \q(t,x_0)$ with initial conditions
$x_0\in\SuE=H^{-1}(E)$ coincide, up to time parametrization, with the
geodesics in the Jacobi metric on $M=\bR^d_\q$
\beq
g_E(\q):= (E-V(\q))\cdot g(\q)\qquad (\q\in M),
\Leq{Jac}
which is conformally equivalent to the Euclidean metric $g$.
For $\htheta\in S^{d-1}$ we consider the Lagrange submanifolds
\[ L_\htheta:= \{x\in s_E\mid \hat{p}^-(x) = \htheta\}.\]
If the potential $V$ is constant, then the
particle has constant momentum. In that case, every energy shell
$\SuE$, $E>V$, has the form of a principal bundle
$\pi:\SuE\ar B\cong S^{d-1}$ with base space $B$ diffeomorphic to the
$(d-1)$-dimensional sphere of directions.
Furthermore, every invariant Lagrange submanifold
$L_\htheta=\pi^{-1}(\htheta)\subset \SuE$,
$\htheta\in B$, projects diffeomorphically to the configuration space
$M$ under the restriction $\tau_\htheta$ of
$\tau:\SuE\ar M$ to $L_\htheta$.
Let us now assume that for some potential $V$ and some energy $E > E_\NT$
all Lagrange submanifolds of the energy shell $\SuE$
project diffeomorphically to $M$.
Then we prove that $V\equiv 0$, contradicting the assumption of the theorem.
The metric $g_E$ defines a connection and thus a canonical decomposition
of $T(TM)$ (the space of phase space vectors)
into a horizontal and a vertical subspace:
\[ T_X T M = T_{X,h} TM\oplus T_{X,v} TM\]
for each phase space point $X=(\dot{\q},\q)\in TM$.
Both $T_{X,h} TM$ and $T_{X,v} TM$ are canonically isomorphic to the
$n$-dimensional space $T_q M$.
A vector in $T_{X,v} TM$ varies the velocity of the particle
keeping its position fixed, whereas the horizontal space $T_{X,h} TM$
describes the direction of parallel transport.
Thus any vector $w\in T_XTM$ can be decomposed into its horizontal and
vertical component: $w=w_h+w_v$. The symplectic two-form $\omega$
is described by the formula
\beq
\omega(w^1,w^2)=\langle w^1_h,w^2_v\rangle - \langle w^2_h,w^1_v\rangle
\Leq{om:Rep}
(Prop.\ 3.1.14 of \cite{Kli2}).
Let $\lambda$ be a Lagrangian subspace of $T_X TM$ which
is transversal
to the vertical subspace $T_{X,v}TM$, i.e.
$\lambda\cap T_{X,v}TM= \{0\}$.
Then there exists an operator
\beq
S:T_{X,h}T{ M}\ar T_{X,v}T M
\Leq{OpS}
such that the vertical and horizontal component of any vector
$w=w_h+w_v\in \lambda$ obey the relation
\beq
w_v=Sw_h.
\end{equation}
The symplectic two-form $\omega$ vanishes on $\lambda$.
Therefore by (\ref{om:Rep}),
\[0=\omega(w^1,w^2) = \langle w^1_h,w^2_v\rangle -
\langle w^2_h,w^1_v\rangle
= \langle w^1_h,Sw^2_h\rangle -
\langle w^2_h,Sw^1_h\rangle,\]
i.e., the operator $S$ describing the Lagrangian space $\lambda$
is symmetric.
By assumption no Lagrangian tangent space $\lambda(x)$, $x\in\SuE$, turns vertical.
Hence, using eq.\ (\ref{OpS}), we can describe $\lambda(x)$ by
a symmetric operator $S(x)$.
Let $\Psi_t:T_1 M\ar T_1 M$ denote the geodesic flow
in the unit tangent bundle $T_1 M$ of $(M,g_E)$ and let
$\eta$ be any vector in the tangent space
$T_X T_1 M$ at the point $X=(q,\dot{q})\in T_1 M$
of this energy shell.
Then after time $t$, $X$ has moved to $X_t:=\Psi_t (X)$,
and the vector $\eta$ has moved to $\eta_t:=T\Psi_t(\eta)$
The horizontal part $\eta_{t,h}=Y(t)$ equals a Jacobi field along the
curve $q(t)=\tau\Psi_t(q,\dot{q})$, whose covariant derivative
$\nabla Y(t)=\eta_{t,v}$ equals the vertical part of $\eta$
(Lemma 3.1.17 of \cite{Kli2}).
By definition, a Jacobi field $Y(t)$ satisfies the so-called
{\em Jacobi equation}
\[\nabla^2 Y(t) + R_{X_t} Y(t) = 0 \]
for the {\em curvature operator}
\beq
R_V:T_q M\ar T_q M,\qquad W\mapsto R(W,V)V,
\Leq{curv:op}
$R$ being the Riemann curvature tensor.
Thus we know that
\[\nabla^2 Y(t) = \nabla(S Y(t))
= \l( \nabla S+S^2 \ri) Y(t) = -R_{X_t} Y(t)\]
for all Jacobi fields $Y(t)$. Hence the operator $S$
satisfies the Riccati equation
\begin{equation}
\nabla S + S^2 + R_{X} =0.
\label{Ric}
\end{equation}
By Lemma \ref{lem:decay} below we may
integrate the trace of this equation over the unit tangent bundle
$T_1 M$.
The integral of the covariant derivative $\nabla S$
vanishes, and the integral
of $\mbox{trace}(S^2)$ is positive. Hence
\begin{equation}
\int_{T_1 M} \mbox{trace}(R_{X}) do\, dm \leq 0,
\end{equation}
where we denote by $dm=\sqrt{\det g^J(q)}dq_1\wedge\ldots\wedge dq_n$ the
measure on $M$ and by $do$ the measure on the unit sphere ($\int_{S^{d-1}}do
=\mbox{vol} (S^{d-1})$).
But
\beq
\int_{T_1 M} \mbox{trace}(R_{X}) do\, dm =
\frac{\mbox{vol} (S^{d-1})}{d}\int_M {\cal R}(\q) dm,
\Leq{RR}
with the scalar curvature ${\cal R}$ of the Jacobi metric.
If the particle moves on a plane $M=\bR^2_\q$, then
$\int_{M} {\cal R}(\q) dm=0$ as a consequence of the Gauss-Bonnet formula.
For $d\geq 3$, that equality is wrong in general. But in our
case the Jacobi metric (\ref{Jac}) is conformally flat.
Defining the positive function
\beq
u: M\ar \bR^+ \qmbox{by}u:=(E-V)^{(d-2)/4},
\Leq{u}
the measure $dm$ on $M$ equals
$dm=u^{\frac{2d}{d-2}}dq_1\wedge\ldots\wedge dq_n$.
The scalar curvature ${\cal R}$ equals
\beq
{\cal R}= 4\frac{1-d}{d-2}u^{-\frac{d+2}{d-2}}\Delta u
\Leq{sc:c}
(with the Euclidean Laplacian
$\Delta=\sum_{i=1}^d \frac{\pa^2}{\pa q_i^2}$ on $M$).
Therefore
\beqn
\int_M {\cal R}(\q) dm
&=& -4\frac{d-1}{d-2}\int_M u^{-\frac{d+2}{d-2}}
(\Delta u) u^{\frac{2d}{d-2}}dq_1\wedge\ldots \wedge dq_d \NN\\
&=& -4\frac{d-1}{d-2}\int_M u(\Delta u)
dq_1\wedge\ldots \wedge dq_d \NN\\
&=& + 4\frac{d-1}{d-2}\int_M (\nabla u)(\nabla u)
dq_1\wedge\ldots \wedge dq_d \geq 0.
\label{larger}
\eeqn
Eqs.\ (\ref{RR}) and (\ref{larger}) are compatible with (\ref{Ric}) only if
the potential $V$ is constant, and thus equal to zero.
As we assumed that $V$ is non-vanishing, not all Lagrange manifolds project
diffeomorphically to the configuration space $M$.
But since the Hamiltonian function is a positive quadratic form,
the folds of the Lagrange manifolds over $M$
extend to spatial infinity, see Duistermaat \cite{Du}.
This implies a divergence of the differential cross section.
\hfill $\Box$
%
\begin{remark} {\rm
The above theorem only implies that if {\em all} scattering
is in the forward direction, then the potential vanishes.
In fact, there exist non-zero potentials which give rise to pure forward
scattering in {\em some} directions.
}\end{remark}
In the next lemma we show decay estimates used in the proof of
Theorem~\ref{theo}.
%
\begin{lemma} \label{lem:decay}
For a short range potential $V\in C^\infty(M,\bR)$
with $\alpha= 2(d-1)$ in (\ref{smooth}), an energy $E>\Vmax$ and
a smooth field of symmetric operators
\[T_1M \ni X\mapsto S(X)\in L(T_{X,h}T{ M}, T_{X,v}T M)\]
meeting the Riccati equation (\ref{Ric}) along any geodesic flow line,
\beq
\int_M \l|{\cal R}(\q)\ri| dm \Vmax$
the Jacobi metric $g_E$ is well-defined, and converges at
infinity to a Euclidean metric.
In the case $d>2$, the function $u$ defined in (\ref{u})
is bounded away from zero.
The expression (\ref{sc:c}) for the scalar curvature, together with
(\ref{smooth}) implies ${\cal R}(\q)=\cO(\LA\q\RA^{-\alpha-2})$ with
$\LA \q\RA:=\sqrt{1+\q^{\,2}}$, so that
(\ref{R:decay}) holds true.
The $(d=2)$-dimensional case gives the same decay estimate, since there
\[{\cal R}(\q)=
\frac{(E-V(\q))\Delta V(\q)-\l(\nabla V(\q)\ri)^2}{2(E-V(\q))^3}.\]
Estimating the norm of $S$ is more complicated. The idea is to exploit that $S$
is finite everywhere, and to prove the estimate
\beq
\int_{S_\q^{d-1}} \l\|S^2(\vec{v},\q)\ri\| d\vec{v} =
\cO\l(\LA\q\RA^{-d-\vep}\ri)
\Leq{A}
on the unit sphere over $\q$,
which then implies (\ref{S:decay}).
We first note that the Riccati equation (\ref{Ric}) tends to develop
singularities. More precisely,
if for some $T>0$ and $X\in T_1M$
\beq
\|S(X)\|>2/T\qmbox{and} \|R_{\psi^t(X)}\|\leq 2/T^2\qquad(t\in[-T,T]),
\Leq{non:finite}
then $S(\psi^t(X))$ meeting (\ref{Ric}) cannot be finite in the whole interval
$[-T,T]$.
Namely let $Y(0)$ be a norm one
eigenvector of $S(X)$ with eigenvalue $s(0)=\pm \|S(X)\|$
(such an eigenvector exists since $S(X)$ is symmetric). Then by using time
inversion, if necessary, we may assume that $s(0)=-\|S(X)\|$.
We set $Y(t)=y(t)\hat{Y}(t)$ where the unit vector $\hat{Y}(t)$ is the
parallel transport of $Y(0)$ along the flow line. Then
by assumption (\ref{non:finite}) $s(t):=\dot{y}(t)/y(t)$
meets the scalar inequality $\dot{s}(t)\leq -s^2(t)+2/T^2$, or $u(t):=1/s(t)$
meets
\[\dot{u}(t)\geq 1 -2u^2(t)/T^2\qmbox{,\ and}-T/2\beta>1-1/(d-1)$ and $T=c\cdot |\q|$ in (\ref{S:small})
we get the contribution
\beq
\int_{R(\q_0)} \l\|S^2(\vec{v}_0,\q_0)\ri\| d\vec{v}_0 =
\cO\l(\LA\q_0\RA^{-\beta(d-1)-2}\ri)=
\cO\l(\LA\q_0\RA^{-d-\vep}\ri)
\Leq{R:est}
of $R(\q_0)$ to (\ref{A}).
So let $\vec{v}_0 \in S_{\q_0}^{d-1}\setminus R(\q_0)$ be the initial direction of
$X_0:=(\vec{v}_0,\q_0)\in T_1M$ and $\theta_0:=\sphericalangle(\vec{v}_0,-\q_0)$
so that $\LA \q_0\RA^{-\beta} < \theta_0 < \pi - \LA \q_0\RA^{-\beta}$.
We claim that for $\LA \q_0\RA$ large
\beq
\inf_{t\in\bR} |\q(t,X_0)| \geq \eh |\q_0| |\sin(\theta_0)|.
\Leq{B}
For vanishing potential $V\equiv 0$, we would have motion on straight lines and
thus $\inf_{t\in\bR} |\q(t,X_0)| = |\q_0| |\sin(\theta_0)|$.
Since the flow is reversible, we may assume w.l.o.g.\ that $\theta_0\leq\pi/2$.
We prove (\ref{B}) by a self-consistent estimate for the double cone
\[\cC(X_0) := \l\{ \q\in M\l|\ \q=\q_0\mbox{ or }
\min_{\vec{v}_0'=\pm \vec{v}_0} \sphericalangle(\q-\q_0,\vec{v}_0') <
\eh\LA \q_0\RA^{-\beta} \ri.\ri\}\]
in configuration space $M$ with vertex $\q_0$ and axis $\vec{v}_0$.
Note that for $\LA \q_0\RA$ large
\beq
{\rm dist} (\cC(X_0),\vec{0}) = |\q_0|\sin(\theta_0-\eh\LA \q_0\RA^{-\beta})
\geq |\q_0|^{1-\vep} \theta_0,
\Leq{C}
since $\theta_0> \LA \q_0\RA^{-\beta}$.
We claim that the trajectory stays in the cone for all times in the sense that
\beq
\q(t,X_0)\in\cC(X_0)\qmbox{and} \sphericalangle\l(\dot{\q}(t,X_0),\vec{v}_0\ri) <
\eh\LA \q_0\RA^{-\beta}\qquad(t\in\bR)
\Leq{D}
It suffices to prove the second inequality, since the first follows
from the second and the definition of $\cC(X_0)$.
The geodesic curvature of the trajectory $t\mapsto \q(t,X_0)$ in the Euclidean
metric on $M=\bR^d_\q$ is given by $k(\psi^t(X_0))$, where the phase space
function $k$ is defined in (\ref{geo:curv}).
For $\LA \q_0\RA$ large and as long as $\q(t,X_0)\in\cC(X_0)$, this is bounded
above by
\beqno
\frac{|\nabla V(\q(t,X_0))|}{E} &=&
\cO\l(|\q(t,X_0)|^{-\alpha-1}\ri)\\
&=& \cO\l((|\q_0|^{1-\vep}\theta_0)^{-\alpha-1}\ri) =
\cO\l(|\q_0|^{-2-\vep}\ri)
\eeqno
for $\beta-(1-1/(d-1))>0$ small, using (\ref{smooth}) and (\ref{C}).
Integrating this curvature along a segment of length $T:= 2|\q_0|$, we see that
within that segment the angle between the initial and the actual direction
is of the order
\[\sphericalangle\l(\dot{\q}(t,X_0),\vec{v}_0\ri) = \cO(\LA \q_0\RA^{-1-\vep})\]
which implies (\ref{D}) for the segment.
Moreover at time $T$ the trajectory already passed its (unique) pericentral
point of minimal distance $|\q(t,X_0)|$ from the origin.
An estimate analogous to (\ref{away}) then allows for a similar statement for
all times $t\geq T$ and $t\leq 0$, proving (\ref{D}).
So we may conclude from (\ref{smooth}) and (\ref{C}) that (\ref{curv:op})
meets the estimate
\[ \|R_{\psi^t(X_0)}\| \leq \l({\rm dist} (\cC(X_0),\vec{0})\ri)^{-\alpha-2}\\
= \cO\l(\LA \q_0\RA^{-2d(1-\vep)}\theta_0^{-2d}\ri).\]
By (\ref{S:small}) $\|S^2(X_0)\|= \|S(X_0)\|^2$ is of the same order
so that
\beqno
\int_{S_{\q_0}^{d-1}\setminus R(\q_0)}
\l\|S^2(\vec{v}_0,\q_0)\ri\| d\vec{v}_0 &=&
\cO\l( \LA \q_0\RA^{-2d(1-\vep)}\int_{\LA \q_0\RA^{-\beta}}^{\pi/2}
\theta^{-2d}\ \theta^{d-2}d\theta\ri)\\
&=& \cO\l(\LA\q\RA^{-d-\frac{2}{d-1}+\vep}\ri).
\eeqno
Together with the similar estimate (\ref{R:est}) for $R(\q_0)$
we have thus shown
the decay estimate (\ref{A}).
\hfill $\Box$
%
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